url stringclasses 147 values | commit stringclasses 147 values | file_path stringlengths 7 101 | full_name stringlengths 1 94 | start stringlengths 6 10 | end stringlengths 6 11 | tactic stringlengths 1 11.2k | state_before stringlengths 3 2.09M | state_after stringlengths 6 2.09M | input stringlengths 73 2.09M |
|---|---|---|---|---|---|---|---|---|---|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | linarith | case inr.mpr.inl
x y : β
h : 0 > y
h' : x < y
β’ x < -y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mpr.inl
x y : β
h : 0 > y
h' : x < y
β’ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.lt_abs | [31, 1] | [50, 15] | exact h' | case inr.mpr.inr
x y : β
h : 0 > y
h' : x < -y
β’ x < -y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mpr.inr
x y : β
h : 0 > y
h' : x < -y
β’ x < -y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | rcases le_or_gt 0 x with h | h | x y : β
β’ |x| < y β -y < x β§ x < y | case inl
x y : β
h : 0 β€ x
β’ |x| < y β -y < x β§ x < y
case inr
x y : β
h : 0 > x
β’ |x| < y β -y < x β§ x < y | Please generate a tactic in lean4 to solve the state.
STATE:
x y : β
β’ |x| < y β -y < x β§ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | rw [abs_of_nonneg h] | case inl
x y : β
h : 0 β€ x
β’ |x| < y β -y < x β§ x < y | case inl
x y : β
h : 0 β€ x
β’ x < y β -y < x β§ x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
x y : β
h : 0 β€ x
β’ |x| < y β -y < x β§ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | constructor | case inl
x y : β
h : 0 β€ x
β’ x < y β -y < x β§ x < y | case inl.mp
x y : β
h : 0 β€ x
β’ x < y β -y < x β§ x < y
case inl.mpr
x y : β
h : 0 β€ x
β’ -y < x β§ x < y β x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inl
x y : β
h : 0 β€ x
β’ x < y β -y < x β§ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | . intro h'
rcases h' with β¨h1, h2β©
exact h2 | case inl.mpr
x y : β
h : 0 β€ x
β’ -y < x β§ x < y β x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mpr
x y : β
h : 0 β€ x
β’ -y < x β§ x < y β x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | intro h' | case inl.mp
x y : β
h : 0 β€ x
β’ x < y β -y < x β§ x < y | case inl.mp
x y : β
h : 0 β€ x
h' : x < y
β’ -y < x β§ x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mp
x y : β
h : 0 β€ x
β’ x < y β -y < x β§ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | constructor | case inl.mp
x y : β
h : 0 β€ x
h' : x < y
β’ -y < x β§ x < y | case inl.mp.left
x y : β
h : 0 β€ x
h' : x < y
β’ -y < x
case inl.mp.right
x y : β
h : 0 β€ x
h' : x < y
β’ x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mp
x y : β
h : 0 β€ x
h' : x < y
β’ -y < x β§ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | exact h' | case inl.mp.right
x y : β
h : 0 β€ x
h' : x < y
β’ x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mp.right
x y : β
h : 0 β€ x
h' : x < y
β’ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | linarith | case inl.mp.left
x y : β
h : 0 β€ x
h' : x < y
β’ -y < x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mp.left
x y : β
h : 0 β€ x
h' : x < y
β’ -y < x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | intro h' | case inl.mpr
x y : β
h : 0 β€ x
β’ -y < x β§ x < y β x < y | case inl.mpr
x y : β
h : 0 β€ x
h' : -y < x β§ x < y
β’ x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mpr
x y : β
h : 0 β€ x
β’ -y < x β§ x < y β x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | rcases h' with β¨h1, h2β© | case inl.mpr
x y : β
h : 0 β€ x
h' : -y < x β§ x < y
β’ x < y | case inl.mpr.intro
x y : β
h : 0 β€ x
h1 : -y < x
h2 : x < y
β’ x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mpr
x y : β
h : 0 β€ x
h' : -y < x β§ x < y
β’ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | exact h2 | case inl.mpr.intro
x y : β
h : 0 β€ x
h1 : -y < x
h2 : x < y
β’ x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inl.mpr.intro
x y : β
h : 0 β€ x
h1 : -y < x
h2 : x < y
β’ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | rw [abs_of_neg h] | case inr
x y : β
h : 0 > x
β’ |x| < y β -y < x β§ x < y | case inr
x y : β
h : 0 > x
β’ -x < y β -y < x β§ x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x y : β
h : 0 > x
β’ |x| < y β -y < x β§ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | constructor | case inr
x y : β
h : 0 > x
β’ -x < y β -y < x β§ x < y | case inr.mp
x y : β
h : 0 > x
β’ -x < y β -y < x β§ x < y
case inr.mpr
x y : β
h : 0 > x
β’ -y < x β§ x < y β -x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr
x y : β
h : 0 > x
β’ -x < y β -y < x β§ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | . intro h'
linarith | case inr.mpr
x y : β
h : 0 > x
β’ -y < x β§ x < y β -x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mpr
x y : β
h : 0 > x
β’ -y < x β§ x < y β -x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | intro h' | case inr.mp
x y : β
h : 0 > x
β’ -x < y β -y < x β§ x < y | case inr.mp
x y : β
h : 0 > x
h' : -x < y
β’ -y < x β§ x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mp
x y : β
h : 0 > x
β’ -x < y β -y < x β§ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | constructor | case inr.mp
x y : β
h : 0 > x
h' : -x < y
β’ -y < x β§ x < y | case inr.mp.left
x y : β
h : 0 > x
h' : -x < y
β’ -y < x
case inr.mp.right
x y : β
h : 0 > x
h' : -x < y
β’ x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mp
x y : β
h : 0 > x
h' : -x < y
β’ -y < x β§ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | . linarith | case inr.mp.right
x y : β
h : 0 > x
h' : -x < y
β’ x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mp.right
x y : β
h : 0 > x
h' : -x < y
β’ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | linarith | case inr.mp.left
x y : β
h : 0 > x
h' : -x < y
β’ -y < x | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mp.left
x y : β
h : 0 > x
h' : -x < y
β’ -y < x
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | linarith | case inr.mp.right
x y : β
h : 0 > x
h' : -x < y
β’ x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mp.right
x y : β
h : 0 > x
h' : -x < y
β’ x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | intro h' | case inr.mpr
x y : β
h : 0 > x
β’ -y < x β§ x < y β -x < y | case inr.mpr
x y : β
h : 0 > x
h' : -y < x β§ x < y
β’ -x < y | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mpr
x y : β
h : 0 > x
β’ -y < x β§ x < y β -x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C03_Logic/solutions/Solutions_S05_Disjunction.lean | C03S05.MyAbs.abs_lt | [52, 1] | [70, 15] | linarith | case inr.mpr
x y : β
h : 0 > x
h' : -y < x β§ x < y
β’ -x < y | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case inr.mpr
x y : β
h : 0 > x
h' : -y < x β§ x < y
β’ -x < y
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture3after.lean | Lec3After.my_lemma_ | [37, 1] | [37, 86] | exact add_left_cancel h | a b c : β
h : a + b = a + c
β’ b = c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : β
h : a + b = a + c
β’ b = c
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C07_Hierarchies/S02_Morphisms.lean | map_inv_of_inv | [75, 1] | [77, 61] | rw [β MonoidHomClassβ.map_mul, h, MonoidHomClassβ.map_one] | M N F : Type
instβΒ² : Monoid M
instβΒΉ : Monoid N
instβ : MonoidHomClassβ F M N
f : F
m m' : M
h : m * m' = 1
β’ βf m * βf m' = 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
M N F : Type
instβΒ² : Monoid M
instβΒΉ : Monoid N
instβ : MonoidHomClassβ F M N
f : F
m m' : M
h : m * m' = 1
β’ βf m * βf m' = 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture5.lean | sequentialLimit_unique | [128, 1] | [150, 2] | intro hl hl' | u : β β β
l l' : β
β’ SequentialLimit u l β SequentialLimit u l' β l = l' | u : β β β
l l' : β
hl : SequentialLimit u l
hl' : SequentialLimit u l'
β’ l = l' | Please generate a tactic in lean4 to solve the state.
STATE:
u : β β β
l l' : β
β’ SequentialLimit u l β SequentialLimit u l' β l = l'
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture5.lean | sequentialLimit_unique | [128, 1] | [150, 2] | by_contra hll' | u : β β β
l l' : β
hl : SequentialLimit u l
hl' : SequentialLimit u l'
β’ l = l' | u : β β β
l l' : β
hl : SequentialLimit u l
hl' : SequentialLimit u l'
hll' : Β¬l = l'
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
u : β β β
l l' : β
hl : SequentialLimit u l
hl' : SequentialLimit u l'
β’ l = l'
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture5.lean | sequentialLimit_unique | [128, 1] | [150, 2] | have : |l - l'| > 0 | u : β β β
l l' : β
hl : SequentialLimit u l
hl' : SequentialLimit u l'
hll' : Β¬l = l'
β’ False | case this
u : β β β
l l' : β
hl : SequentialLimit u l
hl' : SequentialLimit u l'
hll' : Β¬l = l'
β’ |l - l'| > 0
u : β β β
l l' : β
hl : SequentialLimit u l
hl' : SequentialLimit u l'
hll' : Β¬l = l'
this : |l - l'| > 0
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
u : β β β
l l' : β
hl : SequentialLimit u l
hl' : SequentialLimit u l'
hll' : Β¬l = l'
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture5.lean | sequentialLimit_unique | [128, 1] | [150, 2] | rw [SequentialLimit] at hl hl' | u : β β β
l l' : β
hl : SequentialLimit u l
hl' : SequentialLimit u l'
hll' : Β¬l = l'
this : |l - l'| > 0
β’ False | u : β β β
l l' : β
hl : β Ξ΅ > 0, β N, β n β₯ N, |u n - l| < Ξ΅
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
u : β β β
l l' : β
hl : SequentialLimit u l
hl' : SequentialLimit u l'
hll' : Β¬l = l'
this : |l - l'| > 0
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture5.lean | sequentialLimit_unique | [128, 1] | [150, 2] | specialize hl (|l - l'| / 2) (by linarith) | u : β β β
l l' : β
hl : β Ξ΅ > 0, β N, β n β₯ N, |u n - l| < Ξ΅
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
β’ False | u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
hl : β N, β n β₯ N, |u n - l| < |l - l'| / 2
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
u : β β β
l l' : β
hl : β Ξ΅ > 0, β N, β n β₯ N, |u n - l| < Ξ΅
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture5.lean | sequentialLimit_unique | [128, 1] | [150, 2] | obtain β¨N, hNβ© := hl | u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
hl : β N, β n β₯ N, |u n - l| < |l - l'| / 2
β’ False | case intro
u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
N : β
hN : β n β₯ N, |u n - l| < |l - l'| / 2
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
hl : β N, β n β₯ N, |u n - l| < |l - l'| / 2
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture5.lean | sequentialLimit_unique | [128, 1] | [150, 2] | obtain β¨N', hN'β© := hl' (|l - l'| / 2) (by linarith) | case intro
u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
N : β
hN : β n β₯ N, |u n - l| < |l - l'| / 2
β’ False | case intro.intro
u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
N : β
hN : β n β₯ N, |u n - l| < |l - l'| / 2
N' : β
hN' : β n β₯ N', |u n - l'| < |l - l'| / 2
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro
u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
N : β
hN : β n β₯ N, |u n - l| < |l - l'| / 2
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture5.lean | sequentialLimit_unique | [128, 1] | [150, 2] | let Nβ := max N N' | case intro.intro
u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
N : β
hN : β n β₯ N, |u n - l| < |l - l'| / 2
N' : β
hN' : β n β₯ N', |u n - l'| < |l - l'| / 2
β’ False | case intro.intro
u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
N : β
hN : β n β₯ N, |u n - l| < |l - l'| / 2
N' : β
hN' : β n β₯ N', |u n - l'| < |l - l'| / 2
Nβ : β := max N N'
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
N : β
hN : β n β₯ N, |u n - l| < |l - l'| / 2
N' : β
hN' : β n β₯ N', |u n - l'| < |l - l'| / 2
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture5.lean | sequentialLimit_unique | [128, 1] | [150, 2] | specialize hN Nβ (Nat.le_max_left N N') | case intro.intro
u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
N : β
hN : β n β₯ N, |u n - l| < |l - l'| / 2
N' : β
hN' : β n β₯ N', |u n - l'| < |l - l'| / 2
Nβ : β := max N N'
β’ False | case intro.intro
u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
N N' : β
hN' : β n β₯ N', |u n - l'| < |l - l'| / 2
Nβ : β := max N N'
hN : |u Nβ - l| < |l - l'| / 2
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
N : β
hN : β n β₯ N, |u n - l| < |l - l'| / 2
N' : β
hN' : β n β₯ N', |u n - l'| < |l - l'| / 2
Nβ : β := max N N'
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture5.lean | sequentialLimit_unique | [128, 1] | [150, 2] | specialize hN' Nβ (Nat.le_max_right N N') | case intro.intro
u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
N N' : β
hN' : β n β₯ N', |u n - l'| < |l - l'| / 2
Nβ : β := max N N'
hN : |u Nβ - l| < |l - l'| / 2
β’ False | case intro.intro
u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
N N' : β
Nβ : β := max N N'
hN : |u Nβ - l| < |l - l'| / 2
hN' : |u Nβ - l'| < |l - l'| / 2
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
N N' : β
hN' : β n β₯ N', |u n - l'| < |l - l'| / 2
Nβ : β := max N N'
hN : |u Nβ - l| < |l - l'| / 2
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture5.lean | sequentialLimit_unique | [128, 1] | [150, 2] | have : |l - l'| < |l - l'| := by
calc |l - l'|
= |l - u Nβ + (u Nβ - l')| := by ring
_ β€ |l - u Nβ| + |u Nβ - l'| := by exact abs_add (l - u Nβ) (u Nβ - l')
_ = |u Nβ - l| + |u Nβ - l'| := by rw [abs_sub_comm]
_ < |l - l'| := by linarith | case intro.intro
u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
N N' : β
Nβ : β := max N N'
hN : |u Nβ - l| < |l - l'| / 2
hN' : |u Nβ - l'| < |l - l'| / 2
β’ False | case intro.intro
u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
thisβ : |l - l'| > 0
N N' : β
Nβ : β := max N N'
hN : |u Nβ - l| < |l - l'| / 2
hN' : |u Nβ - l'| < |l - l'| / 2
this : |l - l'| < |l - l'|
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
N N' : β
Nβ : β := max N N'
hN : |u Nβ - l| < |l - l'| / 2
hN' : |u Nβ - l'| < |l - l'| / 2
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture5.lean | sequentialLimit_unique | [128, 1] | [150, 2] | linarith | case intro.intro
u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
thisβ : |l - l'| > 0
N N' : β
Nβ : β := max N N'
hN : |u Nβ - l| < |l - l'| / 2
hN' : |u Nβ - l'| < |l - l'| / 2
this : |l - l'| < |l - l'|
β’ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
thisβ : |l - l'| > 0
N N' : β
Nβ : β := max N N'
hN : |u Nβ - l| < |l - l'| / 2
hN' : |u Nβ - l'| < |l - l'| / 2
this : |l - l'| < |l - l'|
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture5.lean | sequentialLimit_unique | [128, 1] | [150, 2] | apply abs_pos.2 | case this
u : β β β
l l' : β
hl : SequentialLimit u l
hl' : SequentialLimit u l'
hll' : Β¬l = l'
β’ |l - l'| > 0 | case this
u : β β β
l l' : β
hl : SequentialLimit u l
hl' : SequentialLimit u l'
hll' : Β¬l = l'
β’ l - l' β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
case this
u : β β β
l l' : β
hl : SequentialLimit u l
hl' : SequentialLimit u l'
hll' : Β¬l = l'
β’ |l - l'| > 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture5.lean | sequentialLimit_unique | [128, 1] | [150, 2] | apply sub_ne_zero.2 | case this
u : β β β
l l' : β
hl : SequentialLimit u l
hl' : SequentialLimit u l'
hll' : Β¬l = l'
β’ l - l' β 0 | case this
u : β β β
l l' : β
hl : SequentialLimit u l
hl' : SequentialLimit u l'
hll' : Β¬l = l'
β’ l β l' | Please generate a tactic in lean4 to solve the state.
STATE:
case this
u : β β β
l l' : β
hl : SequentialLimit u l
hl' : SequentialLimit u l'
hll' : Β¬l = l'
β’ l - l' β 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture5.lean | sequentialLimit_unique | [128, 1] | [150, 2] | exact hll' | case this
u : β β β
l l' : β
hl : SequentialLimit u l
hl' : SequentialLimit u l'
hll' : Β¬l = l'
β’ l β l' | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case this
u : β β β
l l' : β
hl : SequentialLimit u l
hl' : SequentialLimit u l'
hll' : Β¬l = l'
β’ l β l'
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture5.lean | sequentialLimit_unique | [128, 1] | [150, 2] | linarith | u : β β β
l l' : β
hl : β Ξ΅ > 0, β N, β n β₯ N, |u n - l| < Ξ΅
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
β’ |l - l'| / 2 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
u : β β β
l l' : β
hl : β Ξ΅ > 0, β N, β n β₯ N, |u n - l| < Ξ΅
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
β’ |l - l'| / 2 > 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture5.lean | sequentialLimit_unique | [128, 1] | [150, 2] | linarith | u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
N : β
hN : β n β₯ N, |u n - l| < |l - l'| / 2
β’ |l - l'| / 2 > 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
N : β
hN : β n β₯ N, |u n - l| < |l - l'| / 2
β’ |l - l'| / 2 > 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture5.lean | sequentialLimit_unique | [128, 1] | [150, 2] | calc |l - l'|
= |l - u Nβ + (u Nβ - l')| := by ring
_ β€ |l - u Nβ| + |u Nβ - l'| := by exact abs_add (l - u Nβ) (u Nβ - l')
_ = |u Nβ - l| + |u Nβ - l'| := by rw [abs_sub_comm]
_ < |l - l'| := by linarith | u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
N N' : β
Nβ : β := max N N'
hN : |u Nβ - l| < |l - l'| / 2
hN' : |u Nβ - l'| < |l - l'| / 2
β’ |l - l'| < |l - l'| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
N N' : β
Nβ : β := max N N'
hN : |u Nβ - l| < |l - l'| / 2
hN' : |u Nβ - l'| < |l - l'| / 2
β’ |l - l'| < |l - l'|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture5.lean | sequentialLimit_unique | [128, 1] | [150, 2] | ring | u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
N N' : β
Nβ : β := max N N'
hN : |u Nβ - l| < |l - l'| / 2
hN' : |u Nβ - l'| < |l - l'| / 2
β’ |l - l'| = |l - u Nβ + (u Nβ - l')| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
N N' : β
Nβ : β := max N N'
hN : |u Nβ - l| < |l - l'| / 2
hN' : |u Nβ - l'| < |l - l'| / 2
β’ |l - l'| = |l - u Nβ + (u Nβ - l')|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture5.lean | sequentialLimit_unique | [128, 1] | [150, 2] | exact abs_add (l - u Nβ) (u Nβ - l') | u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
N N' : β
Nβ : β := max N N'
hN : |u Nβ - l| < |l - l'| / 2
hN' : |u Nβ - l'| < |l - l'| / 2
β’ |l - u Nβ + (u Nβ - l')| β€ |l - u Nβ| + |u Nβ - l'| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
N N' : β
Nβ : β := max N N'
hN : |u Nβ - l| < |l - l'| / 2
hN' : |u Nβ - l'| < |l - l'| / 2
β’ |l - u Nβ + (u Nβ - l')| β€ |l - u Nβ| + |u Nβ - l'|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture5.lean | sequentialLimit_unique | [128, 1] | [150, 2] | rw [abs_sub_comm] | u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
N N' : β
Nβ : β := max N N'
hN : |u Nβ - l| < |l - l'| / 2
hN' : |u Nβ - l'| < |l - l'| / 2
β’ |l - u Nβ| + |u Nβ - l'| = |u Nβ - l| + |u Nβ - l'| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
N N' : β
Nβ : β := max N N'
hN : |u Nβ - l| < |l - l'| / 2
hN' : |u Nβ - l'| < |l - l'| / 2
β’ |l - u Nβ| + |u Nβ - l'| = |u Nβ - l| + |u Nβ - l'|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture5.lean | sequentialLimit_unique | [128, 1] | [150, 2] | linarith | u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
N N' : β
Nβ : β := max N N'
hN : |u Nβ - l| < |l - l'| / 2
hN' : |u Nβ - l'| < |l - l'| / 2
β’ |u Nβ - l| + |u Nβ - l'| < |l - l'| | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
u : β β β
l l' : β
hl' : β Ξ΅ > 0, β N, β n β₯ N, |u n - l'| < Ξ΅
hll' : Β¬l = l'
this : |l - l'| > 0
N N' : β
Nβ : β := max N N'
hN : |u Nβ - l| < |l - l'| / 2
hN' : |u Nβ - l'| < |l - l'| / 2
β’ |u Nβ - l| + |u Nβ - l'| < |l - l'|
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture5.lean | convergesTo_const | [161, 1] | [161, 80] | sorry | a : β
β’ SequentialLimit (fun n => a) a | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a : β
β’ SequentialLimit (fun n => a) a
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture5.lean | SequentialLimit.add | [164, 1] | [166, 60] | sorry | s t : β β β
a b : β
hs : SequentialLimit s a
ht : SequentialLimit t b
β’ SequentialLimit (fun n => s n + t n) (a + b) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
s t : β β β
a b : β
hs : SequentialLimit s a
ht : SequentialLimit t b
β’ SequentialLimit (fun n => s n + t n) (a + b)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture2.lean | inverse_of_a_commutator | [93, 1] | [102, 2] | rw [commutatorElement_def, commutatorElement_def] | G : Type u_1
instβ : Group G
g h : G
β’ β
g, hββ»ΒΉ = β
h, gβ | G : Type u_1
instβ : Group G
g h : G
β’ (g * h * gβ»ΒΉ * hβ»ΒΉ)β»ΒΉ = h * g * hβ»ΒΉ * gβ»ΒΉ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
instβ : Group G
g h : G
β’ β
g, hββ»ΒΉ = β
h, gβ
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture2.lean | inverse_of_a_commutator | [93, 1] | [102, 2] | rw [mul_inv_rev (g * h * gβ»ΒΉ) hβ»ΒΉ] | G : Type u_1
instβ : Group G
g h : G
β’ (g * h * gβ»ΒΉ * hβ»ΒΉ)β»ΒΉ = h * g * hβ»ΒΉ * gβ»ΒΉ | G : Type u_1
instβ : Group G
g h : G
β’ hβ»ΒΉβ»ΒΉ * (g * h * gβ»ΒΉ)β»ΒΉ = h * g * hβ»ΒΉ * gβ»ΒΉ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
instβ : Group G
g h : G
β’ (g * h * gβ»ΒΉ * hβ»ΒΉ)β»ΒΉ = h * g * hβ»ΒΉ * gβ»ΒΉ
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture2.lean | inverse_of_a_commutator | [93, 1] | [102, 2] | rw [mul_inv_rev] | G : Type u_1
instβ : Group G
g h : G
β’ hβ»ΒΉβ»ΒΉ * (g * h * gβ»ΒΉ)β»ΒΉ = h * g * hβ»ΒΉ * gβ»ΒΉ | G : Type u_1
instβ : Group G
g h : G
β’ hβ»ΒΉβ»ΒΉ * (gβ»ΒΉβ»ΒΉ * (g * h)β»ΒΉ) = h * g * hβ»ΒΉ * gβ»ΒΉ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
instβ : Group G
g h : G
β’ hβ»ΒΉβ»ΒΉ * (g * h * gβ»ΒΉ)β»ΒΉ = h * g * hβ»ΒΉ * gβ»ΒΉ
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture2.lean | inverse_of_a_commutator | [93, 1] | [102, 2] | rw [mul_inv_rev] | G : Type u_1
instβ : Group G
g h : G
β’ hβ»ΒΉβ»ΒΉ * (gβ»ΒΉβ»ΒΉ * (g * h)β»ΒΉ) = h * g * hβ»ΒΉ * gβ»ΒΉ | G : Type u_1
instβ : Group G
g h : G
β’ hβ»ΒΉβ»ΒΉ * (gβ»ΒΉβ»ΒΉ * (hβ»ΒΉ * gβ»ΒΉ)) = h * g * hβ»ΒΉ * gβ»ΒΉ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
instβ : Group G
g h : G
β’ hβ»ΒΉβ»ΒΉ * (gβ»ΒΉβ»ΒΉ * (g * h)β»ΒΉ) = h * g * hβ»ΒΉ * gβ»ΒΉ
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture2.lean | inverse_of_a_commutator | [93, 1] | [102, 2] | rw [inv_inv, inv_inv] | G : Type u_1
instβ : Group G
g h : G
β’ hβ»ΒΉβ»ΒΉ * (gβ»ΒΉβ»ΒΉ * (hβ»ΒΉ * gβ»ΒΉ)) = h * g * hβ»ΒΉ * gβ»ΒΉ | G : Type u_1
instβ : Group G
g h : G
β’ h * (g * (hβ»ΒΉ * gβ»ΒΉ)) = h * g * hβ»ΒΉ * gβ»ΒΉ | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
instβ : Group G
g h : G
β’ hβ»ΒΉβ»ΒΉ * (gβ»ΒΉβ»ΒΉ * (hβ»ΒΉ * gβ»ΒΉ)) = h * g * hβ»ΒΉ * gβ»ΒΉ
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture2.lean | inverse_of_a_commutator | [93, 1] | [102, 2] | rw [mul_assoc] | G : Type u_1
instβ : Group G
g h : G
β’ h * (g * (hβ»ΒΉ * gβ»ΒΉ)) = h * g * hβ»ΒΉ * gβ»ΒΉ | G : Type u_1
instβ : Group G
g h : G
β’ h * (g * (hβ»ΒΉ * gβ»ΒΉ)) = h * g * (hβ»ΒΉ * gβ»ΒΉ) | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
instβ : Group G
g h : G
β’ h * (g * (hβ»ΒΉ * gβ»ΒΉ)) = h * g * hβ»ΒΉ * gβ»ΒΉ
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture2.lean | inverse_of_a_commutator | [93, 1] | [102, 2] | rw [mul_assoc] | G : Type u_1
instβ : Group G
g h : G
β’ h * (g * (hβ»ΒΉ * gβ»ΒΉ)) = h * g * (hβ»ΒΉ * gβ»ΒΉ) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
G : Type u_1
instβ : Group G
g h : G
β’ h * (g * (hβ»ΒΉ * gβ»ΒΉ)) = h * g * (hβ»ΒΉ * gβ»ΒΉ)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture16Before.lean | my_equality | [31, 1] | [35, 8] | have c := a + b | a b : β
β’ a + b + a β€ 2 * (a + b) | a b c : β
β’ a + b + a β€ 2 * (a + b) | Please generate a tactic in lean4 to solve the state.
STATE:
a b : β
β’ a + b + a β€ 2 * (a + b)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture16Before.lean | my_equality | [31, 1] | [35, 8] | have h : c = a + b := sorry | a b c : β
β’ a + b + a β€ 2 * (a + b) | a b c : β
h : c = a + b
β’ a + b + a β€ 2 * (a + b) | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : β
β’ a + b + a β€ 2 * (a + b)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture16Before.lean | my_equality | [31, 1] | [35, 8] | rw [β h] | a b c : β
h : c = a + b
β’ a + b + a β€ 2 * (a + b) | a b c : β
h : c = a + b
β’ c + a β€ 2 * c | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : β
h : c = a + b
β’ a + b + a β€ 2 * (a + b)
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture16Before.lean | my_equality | [31, 1] | [35, 8] | sorry | a b c : β
h : c = a + b
β’ c + a β€ 2 * c | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
a b c : β
h : c = a + b
β’ c + a β€ 2 * c
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Lectures/Lecture16Before.lean | my_lemma | [119, 1] | [121, 53] | rw [Nat.add_succ, my_lemma n] | n : β
β’ 0 + Nat.succ n = Nat.succ n + 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
β’ 0 + Nat.succ n = Nat.succ n + 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Common.lean | pow_self_ne_zero | [151, 1] | [154, 15] | by_cases hn : n = 0 | n : β
β’ n ^ n β 0 | case pos
n : β
hn : n = 0
β’ n ^ n β 0
case neg
n : β
hn : Β¬n = 0
β’ n ^ n β 0 | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
β’ n ^ n β 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Common.lean | pow_self_ne_zero | [151, 1] | [154, 15] | simp [hn] | case pos
n : β
hn : n = 0
β’ n ^ n β 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
n : β
hn : n = 0
β’ n ^ n β 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/Common.lean | pow_self_ne_zero | [151, 1] | [154, 15] | positivity | case neg
n : β
hn : Β¬n = 0
β’ n ^ n β 0 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
n : β
hn : Β¬n = 0
β’ n ^ n β 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.two_le | [9, 1] | [14, 18] | cases m | m : β
h0 : m β 0
h1 : m β 1
β’ 2 β€ m | case zero
h0 : 0 β 0
h1 : 0 β 1
β’ 2 β€ 0
case succ
nβ : β
h0 : nβ + 1 β 0
h1 : nβ + 1 β 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1
β’ 2 β€ nβ + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
m : β
h0 : m β 0
h1 : m β 1
β’ 2 β€ m
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.two_le | [9, 1] | [14, 18] | contradiction | case zero
h0 : 0 β 0
h1 : 0 β 1
β’ 2 β€ 0
case succ
nβ : β
h0 : nβ + 1 β 0
h1 : nβ + 1 β 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1
β’ 2 β€ nβ + 1 | case succ
nβ : β
h0 : nβ + 1 β 0
h1 : nβ + 1 β 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1
β’ 2 β€ nβ + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
h0 : 0 β 0
h1 : 0 β 1
β’ 2 β€ 0
case succ
nβ : β
h0 : nβ + 1 β 0
h1 : nβ + 1 β 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1
β’ 2 β€ nβ + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.two_le | [9, 1] | [14, 18] | case succ m hm =>
cases m; contradiction
repeat' apply Nat.succ_le_succ
apply zero_le | m : β
h0 : m + 1 β 0
h1 : m + 1 β 1
hm : m + 1 = m β 2 β€ m + 1
β’ 2 β€ m + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
m : β
h0 : m + 1 β 0
h1 : m + 1 β 1
hm : m + 1 = m β 2 β€ m + 1
β’ 2 β€ m + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.two_le | [9, 1] | [14, 18] | cases m | m : β
h0 : m + 1 β 0
h1 : m + 1 β 1
hm : m + 1 = m β 2 β€ m + 1
β’ 2 β€ m + 1 | case zero
h0 : 0 + 1 β 0
h1 : 0 + 1 β 1
hm : 0 + 1 = 0 β 2 β€ 0 + 1
β’ 2 β€ 0 + 1
case succ
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
hm : nβ + 1 + 1 = nβ + 1 β 2 β€ nβ + 1 + 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1 + 1
β’ 2 β€ nβ + 1 + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
m : β
h0 : m + 1 β 0
h1 : m + 1 β 1
hm : m + 1 = m β 2 β€ m + 1
β’ 2 β€ m + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.two_le | [9, 1] | [14, 18] | contradiction | case zero
h0 : 0 + 1 β 0
h1 : 0 + 1 β 1
hm : 0 + 1 = 0 β 2 β€ 0 + 1
β’ 2 β€ 0 + 1
case succ
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
hm : nβ + 1 + 1 = nβ + 1 β 2 β€ nβ + 1 + 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1 + 1
β’ 2 β€ nβ + 1 + 1 | case succ
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
hm : nβ + 1 + 1 = nβ + 1 β 2 β€ nβ + 1 + 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1 + 1
β’ 2 β€ nβ + 1 + 1 | Please generate a tactic in lean4 to solve the state.
STATE:
case zero
h0 : 0 + 1 β 0
h1 : 0 + 1 β 1
hm : 0 + 1 = 0 β 2 β€ 0 + 1
β’ 2 β€ 0 + 1
case succ
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
hm : nβ + 1 + 1 = nβ + 1 β 2 β€ nβ + 1 + 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1 + 1
β’ 2 β€ nβ + 1 + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.two_le | [9, 1] | [14, 18] | repeat' apply Nat.succ_le_succ | case succ
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
hm : nβ + 1 + 1 = nβ + 1 β 2 β€ nβ + 1 + 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1 + 1
β’ 2 β€ nβ + 1 + 1 | case succ.a.a
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
hm : nβ + 1 + 1 = nβ + 1 β 2 β€ nβ + 1 + 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1 + 1
β’ 0 β€ nβ | Please generate a tactic in lean4 to solve the state.
STATE:
case succ
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
hm : nβ + 1 + 1 = nβ + 1 β 2 β€ nβ + 1 + 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1 + 1
β’ 2 β€ nβ + 1 + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.two_le | [9, 1] | [14, 18] | apply zero_le | case succ.a.a
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
hm : nβ + 1 + 1 = nβ + 1 β 2 β€ nβ + 1 + 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1 + 1
β’ 0 β€ nβ | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.a.a
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
hm : nβ + 1 + 1 = nβ + 1 β 2 β€ nβ + 1 + 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1 + 1
β’ 0 β€ nβ
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.two_le | [9, 1] | [14, 18] | apply Nat.succ_le_succ | case succ.a
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
hm : nβ + 1 + 1 = nβ + 1 β 2 β€ nβ + 1 + 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1 + 1
β’ 1 β€ nβ + 1 | case succ.a.a
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
hm : nβ + 1 + 1 = nβ + 1 β 2 β€ nβ + 1 + 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1 + 1
β’ 0 β€ nβ | Please generate a tactic in lean4 to solve the state.
STATE:
case succ.a
nβ : β
h0 : nβ + 1 + 1 β 0
h1 : nβ + 1 + 1 β 1
hm : nβ + 1 + 1 = nβ + 1 β 2 β€ nβ + 1 + 1
aβ : nβ + 1 = nβ β 2 β€ nβ + 1 + 1
β’ 1 β€ nβ + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [28, 1] | [44, 27] | by_cases np : n.Prime | n : β
h : 2 β€ n
β’ β p, Nat.Prime p β§ p β£ n | case pos
n : β
h : 2 β€ n
np : Nat.Prime n
β’ β p, Nat.Prime p β§ p β£ n
case neg
n : β
h : 2 β€ n
np : Β¬Nat.Prime n
β’ β p, Nat.Prime p β§ p β£ n | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
h : 2 β€ n
β’ β p, Nat.Prime p β§ p β£ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [28, 1] | [44, 27] | induction' n using Nat.strong_induction_on with n ih | case neg
n : β
h : 2 β€ n
np : Β¬Nat.Prime n
β’ β p, Nat.Prime p β§ p β£ n | case neg.h
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : Β¬Nat.Prime n
β’ β p, Nat.Prime p β§ p β£ n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
n : β
h : 2 β€ n
np : Β¬Nat.Prime n
β’ β p, Nat.Prime p β§ p β£ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [28, 1] | [44, 27] | rw [Nat.prime_def_lt] at np | case neg.h
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : Β¬Nat.Prime n
β’ β p, Nat.Prime p β§ p β£ n | case neg.h
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : Β¬(2 β€ n β§ β m < n, m β£ n β m = 1)
β’ β p, Nat.Prime p β§ p β£ n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : Β¬Nat.Prime n
β’ β p, Nat.Prime p β§ p β£ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [28, 1] | [44, 27] | push_neg at np | case neg.h
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : Β¬(2 β€ n β§ β m < n, m β£ n β m = 1)
β’ β p, Nat.Prime p β§ p β£ n | case neg.h
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
β’ β p, Nat.Prime p β§ p β£ n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : Β¬(2 β€ n β§ β m < n, m β£ n β m = 1)
β’ β p, Nat.Prime p β§ p β£ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [28, 1] | [44, 27] | rcases np h with β¨m, mltn, mdvdn, mne1β© | case neg.h
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
β’ β p, Nat.Prime p β§ p β£ n | case neg.h.intro.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
β’ β p, Nat.Prime p β§ p β£ n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
β’ β p, Nat.Prime p β§ p β£ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [28, 1] | [44, 27] | have : m β 0 := by
intro mz
rw [mz, zero_dvd_iff] at mdvdn
linarith | case neg.h.intro.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
β’ β p, Nat.Prime p β§ p β£ n | case neg.h.intro.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
β’ β p, Nat.Prime p β§ p β£ n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.intro.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
β’ β p, Nat.Prime p β§ p β£ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [28, 1] | [44, 27] | have mgt2 : 2 β€ m := two_le this mne1 | case neg.h.intro.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
β’ β p, Nat.Prime p β§ p β£ n | case neg.h.intro.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
β’ β p, Nat.Prime p β§ p β£ n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.intro.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
β’ β p, Nat.Prime p β§ p β£ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [28, 1] | [44, 27] | by_cases mp : m.Prime | case neg.h.intro.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
β’ β p, Nat.Prime p β§ p β£ n | case pos
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Nat.Prime m
β’ β p, Nat.Prime p β§ p β£ n
case neg
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬Nat.Prime m
β’ β p, Nat.Prime p β§ p β£ n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.h.intro.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
β’ β p, Nat.Prime p β§ p β£ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [28, 1] | [44, 27] | . rcases ih m mltn mgt2 mp with β¨p, pp, pdvdβ©
use p, pp
apply pdvd.trans mdvdn | case neg
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬Nat.Prime m
β’ β p, Nat.Prime p β§ p β£ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬Nat.Prime m
β’ β p, Nat.Prime p β§ p β£ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [28, 1] | [44, 27] | use n, np | case pos
n : β
h : 2 β€ n
np : Nat.Prime n
β’ β p, Nat.Prime p β§ p β£ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
n : β
h : 2 β€ n
np : Nat.Prime n
β’ β p, Nat.Prime p β§ p β£ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [28, 1] | [44, 27] | intro mz | n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
β’ m β 0 | n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
mz : m = 0
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
β’ m β 0
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [28, 1] | [44, 27] | rw [mz, zero_dvd_iff] at mdvdn | n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
mz : m = 0
β’ False | n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : n = 0
mne1 : m β 1
mz : m = 0
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
mz : m = 0
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [28, 1] | [44, 27] | linarith | n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : n = 0
mne1 : m β 1
mz : m = 0
β’ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : n = 0
mne1 : m β 1
mz : m = 0
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [28, 1] | [44, 27] | use m, mp | case pos
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Nat.Prime m
β’ β p, Nat.Prime p β§ p β£ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case pos
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Nat.Prime m
β’ β p, Nat.Prime p β§ p β£ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [28, 1] | [44, 27] | rcases ih m mltn mgt2 mp with β¨p, pp, pdvdβ© | case neg
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬Nat.Prime m
β’ β p, Nat.Prime p β§ p β£ n | case neg.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬Nat.Prime m
p : β
pp : Nat.Prime p
pdvd : p β£ m
β’ β p, Nat.Prime p β§ p β£ n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬Nat.Prime m
β’ β p, Nat.Prime p β§ p β£ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [28, 1] | [44, 27] | use p, pp | case neg.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬Nat.Prime m
p : β
pp : Nat.Prime p
pdvd : p β£ m
β’ β p, Nat.Prime p β§ p β£ n | case right
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬Nat.Prime m
p : β
pp : Nat.Prime p
pdvd : p β£ m
β’ p β£ n | Please generate a tactic in lean4 to solve the state.
STATE:
case neg.intro.intro
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬Nat.Prime m
p : β
pp : Nat.Prime p
pdvd : p β£ m
β’ β p, Nat.Prime p β§ p β£ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.exists_prime_factor | [28, 1] | [44, 27] | apply pdvd.trans mdvdn | case right
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬Nat.Prime m
p : β
pp : Nat.Prime p
pdvd : p β£ m
β’ p β£ n | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case right
n : β
ih : β m < n, 2 β€ m β Β¬Nat.Prime m β β p, Nat.Prime p β§ p β£ m
h : 2 β€ n
np : 2 β€ n β β m < n, m β£ n β§ m β 1
m : β
mltn : m < n
mdvdn : m β£ n
mne1 : m β 1
this : m β 0
mgt2 : 2 β€ m
mp : Β¬Nat.Prime m
p : β
pp : Nat.Prime p
pdvd : p β£ m
β’ p β£ n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [46, 1] | [60, 8] | intro n | β’ β (n : β), β p > n, Nat.Prime p | n : β
β’ β p > n, Nat.Prime p | Please generate a tactic in lean4 to solve the state.
STATE:
β’ β (n : β), β p > n, Nat.Prime p
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [46, 1] | [60, 8] | have : 2 β€ Nat.factorial (n + 1) + 1 := by
sorry | n : β
β’ β p > n, Nat.Prime p | n : β
this : 2 β€ Nat.factorial (n + 1) + 1
β’ β p > n, Nat.Prime p | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
β’ β p > n, Nat.Prime p
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [46, 1] | [60, 8] | rcases exists_prime_factor this with β¨p, pp, pdvdβ© | n : β
this : 2 β€ Nat.factorial (n + 1) + 1
β’ β p > n, Nat.Prime p | case intro.intro
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
β’ β p > n, Nat.Prime p | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
β’ β p > n, Nat.Prime p
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [46, 1] | [60, 8] | refine' β¨p, _, ppβ© | case intro.intro
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
β’ β p > n, Nat.Prime p | case intro.intro
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
β’ p > n | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
β’ β p > n, Nat.Prime p
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [46, 1] | [60, 8] | by_contra ple | case intro.intro
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
β’ p > n | case intro.intro
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : Β¬p > n
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
β’ p > n
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [46, 1] | [60, 8] | push_neg at ple | case intro.intro
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : Β¬p > n
β’ False | case intro.intro
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : Β¬p > n
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [46, 1] | [60, 8] | have : p β£ Nat.factorial (n + 1) := by
sorry | case intro.intro
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
β’ False | case intro.intro
n : β
thisβ : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
this : p β£ Nat.factorial (n + 1)
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [46, 1] | [60, 8] | have : p β£ 1 := by
sorry | case intro.intro
n : β
thisβ : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
this : p β£ Nat.factorial (n + 1)
β’ False | case intro.intro
n : β
thisβΒΉ : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
thisβ : p β£ Nat.factorial (n + 1)
this : p β£ 1
β’ False | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
n : β
thisβ : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
this : p β£ Nat.factorial (n + 1)
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [46, 1] | [60, 8] | sorry | case intro.intro
n : β
thisβΒΉ : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
thisβ : p β£ Nat.factorial (n + 1)
this : p β£ 1
β’ False | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
case intro.intro
n : β
thisβΒΉ : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
thisβ : p β£ Nat.factorial (n + 1)
this : p β£ 1
β’ False
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [46, 1] | [60, 8] | sorry | n : β
β’ 2 β€ Nat.factorial (n + 1) + 1 | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
β’ 2 β€ Nat.factorial (n + 1) + 1
TACTIC:
|
https://github.com/fpvandoorn/LeanCourse23.git | 7b0a3cf61b802764dc7baee9d9825e9c62cf9c5d | LeanCourse/MIL/C05_Elementary_Number_Theory/S03_Infinitely_Many_Primes.lean | C05S03.primes_infinite | [46, 1] | [60, 8] | sorry | n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
β’ p β£ Nat.factorial (n + 1) | no goals | Please generate a tactic in lean4 to solve the state.
STATE:
n : β
this : 2 β€ Nat.factorial (n + 1) + 1
p : β
pp : Nat.Prime p
pdvd : p β£ Nat.factorial (n + 1) + 1
ple : p β€ n
β’ p β£ Nat.factorial (n + 1)
TACTIC:
|
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